In mathematics a field of sets is a pair where is a set and is an algebra over i.e., a non-empty subset of the power set of closed under the intersection and union of pairs of sets and under complements of individual sets. In other words forms a subalgebra of the power set Boolean algebra of . (Many authors refer to itself as a field of sets. The word "field" in "field of sets" is not used with the meaning of field from field theory.) Elements of are called points and those of are called complexes.
Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.
Famous quotes containing the words field of, field and/or sets:
“We need a type of theatre which not only releases the feelings, insights and impulses possible within the particular historical field of human relations in which the action takes place, but employs and encourages those thoughts and feelings which help transform the field itself.”
—Bertolt Brecht (1898–1956)
“In the quilts I had found good objects—hospitable, warm, with soft edges yet resistant, with boundaries yet suggesting a continuous safe expanse, a field that could be bundled, a bundle that could be unfurled, portable equipment, light, washable, long-lasting, colorful, versatile, functional and ornamental, private and universal, mine and thine.”
—Radka Donnell-Vogt, U.S. quiltmaker. As quoted in Lives and Works, by Lynn F. Miller and Sally S. Swenson (1981)
“Certain anthropologists hold that man, having discovered tools, ceased to evolve biologically. Animals, never having discovered them, continue to fashion drills out of their beaks, oars out of their hind feet, wings out of their forefeet, suits of armor out of their hides, levers out of their horns, saws out of their teeth. Whether this be true or not, all authorities agree that man is the tool-using animal. It sets him off from the rest of the animal kingdom as drastically as does speech.”
—Stuart Chase (1888–1985)